We can see that the potentials of the scalar Regge-Wheeler and Zerilli equations (see [1]}) decay as \(r^{-3}\) , whence the ones of (REF ) and (REF ) decay as \(r^{-2}\) .
The rescaled equations
In order to obtain the conformal scattering theory for the tensorial Fackerell-Ipser equations (REF ) and (REF ) we need to establish their conformal rescaled forms in the conformal compactification domain \(\bar{\mathcal {B}_I}\) .
Since \(r^2\rho \)  and \(r^2\sigma \)  satisfy the scalar wave equation (REF ) and this equation is conformal invariant under the following formula
\(({\Box }_g + \frac{1}{6}\mathrm {Scal}_g)\psi = \Omega ^3 ({\Box }_{\hat{g}} + \frac{1}{6}\mathrm {Scal}_{\hat{g}}) \Omega ^{-1}\psi ,\) 
then \(r(r^2\rho )\)  and \(r(r^2\sigma )\)  satisfy the rescaled scalar wave equation
\(\frac{r^2}{F}L\underline{L}\hat{\psi }- r^2 {\Delta }\hat{\psi }+ 2MR\hat{\psi }= 0 \hbox{ where } \hat{\psi }= r\psi .\) 
Putting
\(\hat{\phi }_a = r\phi _a = r[\nabla _a r(r^2\rho ) + e_{ab}\nabla ^b r(r^2\sigma )].\) 
We commute (REF ) with the projected covariant angular derivative \(r{\nabla }\)  by the same way in Remark REF , then we establish that \(\hat{\phi }_a\)  is a solution of the following equation
\(\frac{1}{F}{{\nabla }}_{\hat{L}}{{\nabla }}_{\hat{\underline{L}}}\hat{\phi }_a - {\Delta }_{\mathbb {S}^2}\hat{\phi }_a + (2MR+1)\hat{\phi }_a = 0,\) 
where \(\hat{L}= r^2L\) , \(\hat{\underline{L}} = \underline{L}\)  and \({\Delta }_{\mathbb {S}^2}=r^2{\Delta }\) .
Therefore, the tensorial equation (REF ) has the following rescaled form in the compactification domain \(\bar{\mathcal {B}}_I\) :
\({\Box }_{\hat{g}}\hat{\phi }_a + (2MR+1)\hat{\phi }_a = 0,\) 
where
\({\Box }_{\hat{g}} = \frac{1}{F}{{\nabla }}_{\hat{L}} {{\nabla }}_{\hat{\underline{L}}} - {\Delta }_{\mathbb {S}^2}.\) 
By the same way Equation (REF ) has the rescaled form as
\({\Box }_{\hat{g}}\underline{\hat{\phi }}_a + (2MR+1)\underline{\hat{\phi }}_a = 0,\) 
where \(\underline{\hat{\phi }}_a=r \underline{\phi }_a\) .
For the convenience in the next sections we multiply Equations (REF ) and (REF ) by a factor \(FR^2\)  and we get the following forms respectively
\({{\nabla }}_{L} {{\nabla }}_{{\underline{L}}}\hat{\phi }_a - FR^2{\Delta }_{\mathbb {S}^2}\hat{\phi }_a + (2MR+1)FR^2\hat{\phi }_a = 0,\) 
\({{\nabla }}_{L} {{\nabla }}_{{\underline{L}}}\underline{\hat{\phi }}_a - FR^2{\Delta }_{\mathbb {S}^2}\hat{\phi }_a + (2MR+1)FR^2\underline{\hat{\phi }}_a = 0.\) 
Remark 2
\(\bullet \)
Since the tensorial Equations (REF ) and (REF ) have the rescaled forms (REF ) and (REF ) respectively, we can see that they are not conformal invariance. The reason is due to the appearance of the potential \(\dfrac{1}{r^2}\phi \)  which arises from the commutator \([r{\nabla },{\Delta }]\) . However, this potential decay is fast enough to ensure that the energy and pointwise decays of the solutions lead to the energy identity up to the future null infinity \(i^+\) . Therefore, we can still obtain conformal scattering theories for Equations (REF ) and (REF ) (see Section 3).
\(\bullet \)
Equations (REF ) and (REF ) (and also the original Equations (REF ) and (REF )) have tensorial forms, precisely 2-forms on the 2-sphere \(\mathbb {S}^2_{(t,r)}\) .
Taking their decompositions on the basic frame \(({\nabla }_a,{\nabla }_b)\)  of \(\Lambda ^1{(\mathcal {B})}\)  we get
\(P\hat{\Phi } + L_1 \hat{\Phi } = 0 \, ,\) 
where
\(P= \left(\begin{matrix}\Box _g &&0\\0&&\Box _g\end{matrix} \right)\) 
is the \(2\times 2\) -matrix diagram with \(\Box _g = \frac{1}{F}L\underline{L} - {\Delta }\) ,
\(\hat{\Phi } = \left( \begin{matrix} \hat{\phi }_1\\ \hat{\phi }_2\end{matrix} \right)\) 
is the components of \(\hat{\phi }_a\)  on the basic frame \(({\nabla }_a,{\nabla }_b)\)  of \(\Lambda ^1(\mathcal {B})\)  and
\(L_1 = \left( \begin{matrix} L_1^{11}&&L_1^{12}\\ L_1^{21}&& L_1^{22} \end{matrix} \right)\) 
is the \(2\times 2\) -matrix where the components are the operators that have the smooth coefficients
\(L_1^{ij} = b_0^{ij}\partial _t + b_a^{ij}\partial _a + c^{ij}.\) 
Energies of the tensorial Fackerell-Ipser field
Energy conservation for the original field
Let \({g}_{ab}\)  be the restriction of the Schwarzschild metric \(g\)  to \(\mathbb {S}^2_{(t,r)}\) . We have \({g}_{ab} = r^2 g_{\mathbb {S}^2}\)  and the following identities hold
\({\nabla }_L{g}^{ab} = {\nabla }_{\underline{L}}{g}^{ab} = 0.\) 
Indeed, by a straightforward calculation we have \({\nabla }_L \partial _{\theta ^a} = FR \partial _{\theta ^a}\)  then
\({\nabla }_L {g}^{ab} = ({\nabla }_L r^{-2} )g_{\mathbb {S}^2}^{-1} + r^2 {\nabla }_Lg_{\mathbb {S}^2}^{-1} = -2FR^3g_{\mathbb {S}^2}^{-1} + 2FR^3g_{\mathbb {S}^2}^{-1} =0. \) 
Therefore, for a 1-form \(\phi _a \in \Lambda ^1(\mathcal {B})\)  on the 2-sphere \(\mathbb {S}^2_{(t,r)}\)  we can define
\({\phi }^a = {\phi }_b{g}^{ab}, \hbox{ } {\nabla }_{\underline{L}}{\phi }^a = {\nabla }_{\underline{L}}{\phi }_b {g}^{ab} \hbox{ and } {\nabla }_L{\phi }^a = {\nabla }_L{\phi }_b {g}^{ab}.\) 
Putting
\(|\Phi |^2 := {\phi }_a {\phi }^a , \, |{\nabla }_{\underline{L}}{\Phi }|^2 := {\nabla }_{\underline{L}}{\phi }_a {\nabla }_{\underline{L}}{\phi }^a \hbox{ and } |{\nabla }_L\Phi |^2 := {\nabla }_L{\phi }_a {\nabla }_L{\phi }^a,\) 
where the scalar product depends on the metric \({g}_{ab}\) .
The Fackerell-Ipser Equation (REF ) leads to
\(&&4\left( {\nabla }_{\underline{L}}{\nabla }_L{\phi }_a - F{\Delta }{\phi }_a + R^2F{\phi }_a \right) {\nabla }_{L}{\phi }^a \\&&+ 4\left( {\nabla }_{\underline{L}}{\nabla }_L{\phi }_a - F{\Delta }{\phi }_a + R^2F{\phi }_a \right) {\nabla }_{\underline{L}}{\phi }^a =0.\) 
This equation is equivalent to
\(&&2{\nabla }_{\underline{L}} \left( |{\nabla }_L{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right)\nonumber \\&&+2{\nabla }_{L} \left( |{\nabla }_{\underline{L}}{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) =0,\) 
here \({\nabla }_{\mathbb {S}^2}\)  is the angular derivative on the unit 2-sphere \(\mathbb {S}^2\) .
Since \({g}^{-1} = 4F^{-1}\partial _u\partial _v - r^{-2}g^{-1}_{\mathbb {S}^2}\)  we have that (REF ) is equivalent to
\(&&8F^{-1}{\nabla }^{L} \left( |{\nabla }_L\hat{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\nonumber \\&&+8F^{-1} {\nabla }^{\underline{L}} \left( |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2} \hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) =0.\) 
Equation (REF ) is also equivalent to
\(({{\nabla }}_L + {\nabla }_{\underline{L}}) \left( |{\nabla }_L{\Phi }|^2 + |{\nabla }_{\underline{L}}{\Phi }|^2 + 2F|R{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) \\- ({\nabla }_L - {\nabla }_{\underline{L}}) \left( |{\nabla }_L{\Phi }|^2 - |{\nabla }_{\underline{L}}{\Phi }|^2 \right) =0\,.\) 
Hence, in the coordinates \((t,\, r_*,\, \omega )\)  Equation (REF ) becomes
\(&&2{\nabla }_{\partial _t} \left( |{\nabla }_L{\Phi }|^2 + |{\nabla }_{\underline{L}}{\Phi }|^2 + 2F|R{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right)\nonumber \\&&+2 {\nabla }_{\partial _{r_*}} \left( |{\nabla }_L{\Phi }|^2 - |{\nabla }_{\underline{L}}{\Phi }|^2 \right) =0.\) 
Since \({g}^{-1} = F^{-1}(\partial _t^2 - \partial _{r_*}^2) - r^{-2}g^{-1}_{\mathbb {S}^2}\) , the above equation can be written as
\(&&2F^{-1}{\nabla }^{\partial _t} \left( |{\nabla }_L{\Phi }|^2 + |{\nabla }_{\underline{L}}{\Phi }|^2 + 2F|R{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right)\nonumber \\&&+2F^{-1} {\nabla }^{\partial _{r_*}} \left( |{\nabla }_L{\Phi }|^2 - |{\nabla }_{\underline{L}}{\Phi }|^2 \right) =0 .\) 
Equations (REF ) and (REF ) are the expressions of the tensorial conservation law for the tensorial Fackerell-Ipser Equation (REF ) in the coordinates \((u,\, v,\, \omega )\)  and \((t,\, r_*,\omega )\)  respectively.
Using (REF ) we can calculate the energy fluxes which across the null hypersurfaces. On the other hand, the energy flux across the spacelike hypersurface such as \(\left\lbrace  t= 0 \right\rbrace \)  can be computed from (REF ). We can re-write (REF ) and (REF ) by a generalized form as
\(g(r,F){\nabla }^a C(\Phi ) = 0,\) 
where \(\partial _a \in \left\lbrace  L,\, \underline{L}, \, \partial _t,\, \partial _{r_*} \right\rbrace \)  and \(g(r,F)=8F^{-1}\)  in coordinates \((u,\,v,\,\omega )\)  and \(g(r,F)=2F^{-1}\)  in coordinates \((t,\, r_*,\, \omega )\) .
Suppose \(\phi _a\)  is a solution of Equation (REF ) with the smooth compactly supported initial data. We integrate the conservation law (REF ) of \({\phi }_a\)  on the domain \(O\)  bounded by the hypersurfaces \(\Sigma _0\)  and \(\Sigma _t\)  in \((\mathcal {B}_I,g)\)  and then by using Stokes's formula we get
\(\int _O g(r,F){\nabla }^a C(\Phi ) \mathrm {dVol}_{g} = \int _{\partial O} C(\Phi ) g(r,F)\partial _a { \begin{picture}(.833,.8)\put (.15,.08){\line (1,0){.35}}\put (.5,.08){\line (0,1){.5}}\end{picture}}\mathrm {dVol}_{g},\) 
where \(\mathrm {dVol}_{g} = r^2F\mathrm {d}t \mathrm {d}r_* \mathrm {d}\mathbb {S}^2\) . This implies that
\(\mathcal {E}_{\Sigma _t}({\Phi }) = \mathcal {E}_{\Sigma _0}({\Phi }),\) 
where
\(\mathcal {E}_{\Sigma _0}(\Phi ) = 4\int _{\Sigma _0}\left( |{\nabla }_{\partial _{t}}{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}r_*\mathrm {d}\mathbb {S}^2\) 
and
\(\mathcal {E}_{\Sigma _t}(\Phi ) = 4\int _{\Sigma _t}\left( |{\nabla }_{\partial _{t}}{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}r_*\mathrm {d}\mathbb {S}^2.\) 
We consider the domain \(\Omega \)  which is bounded by the following five hypersurfaces
\(\bar{\Sigma }_0 = \left\lbrace  t=0,\, u \le u_{2T},\, v\le v_{2T} \right\rbrace ,\, \mathcal {H}^+_{2u_T} = \left\lbrace u=2u_T,v\le v_T \right\rbrace ,\,\mathcal {I}^+_{2v_T}= \left\lbrace v=2v_T,u \le u_T \right\rbrace ,\) 
\(\mathcal {H}_{u_T}= \left\lbrace u=u_T, v_T \le v \le 2v_T \right\rbrace ,\, \mathcal {I}_{v_T}= \left\lbrace v=v_T,u_T \le u \le 2u_T \right\rbrace .\) 
<FIGURE>By integrating the conservation law (REF ) of \({\phi }_a\)  on the domain \(\Omega \)  in \((\mathcal {B}_I,g)\)  and by using again Stokes's formula we get
\(\int _\Omega g(r,F){\nabla }^a C(\Phi ) \mathrm {dVol}_{g} = \int _{\partial \Omega } C(\Phi ) g(r,F)\partial _a { \begin{picture}(.833,.8)\put (.15,.08){\line (1,0){.35}}\put (.5,.08){\line (0,1){.5}}\end{picture}}\mathrm {dVol}_{g},\) 
where \(\mathrm {dVol}_{g} = \dfrac{r^2F}{2}\mathrm {d}u \mathrm {d}v \mathrm {d}\mathbb {S}^2 = r^2F\mathrm {d}t \mathrm {d}r_* \mathrm {d}\mathbb {S}^2\) . This implies that
\(\mathcal {E}_{\mathcal {H}^+_{2u_T}}({\Phi }) + \mathcal {E}_{\mathcal {I}_{v_T}}({\Phi }) + \mathcal {E}_{\mathcal {H}_{u_T}}({\Phi }) + \mathcal {E}_{\mathcal {I}_{2v_T}^+}({\Phi }) = \mathcal {E}_{\bar{\Sigma }_0}(\Phi ),\) 
where
\(\mathcal {E}_{\bar{\Sigma }_0}(\Phi ) = 4\int _{\bar{\Sigma }_0}\left( |{\nabla }_{\partial _{t}}{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}r_*\mathrm {d}\mathbb {S}^2,\) 
\(\mathcal {E}_{\mathcal {I}^+_{2v_T}}({\Phi }) = 4\int _{-\infty }^{u_T} \int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}u \mathrm {d}\mathbb {S}^2,\) 
\(\mathcal {E}_{\mathcal {I}_{v_T}}({\Phi }) = 4\int _{u_T}^{2u_T}\int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}u \mathrm {d}\mathbb {S}^2.\) 
\(\mathcal {E}_{\mathcal {H}^+_{2u_T}}({\Phi }) = 4\int ^{v_T}_{-\infty } \int _{\mathbb {S}^2} \left( |{\nabla }_L{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}v \mathrm {d}\mathbb {S}^2, \) 
\(\mathcal {E}_{\mathcal {H}_{u_T}}({\Phi }) = 4\int ^{2v_T}_{v_T} \int _{\mathbb {S}^2} \left( |{\nabla }_L{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + R^2F|{\Phi }|^2 \right) r^2\mathrm {d}v \mathrm {d}\mathbb {S}^2. \) 

Energy conservation for the rescaled field
Since the rescaled field \(\hat{\phi }_a\)  has the decomposition on the basic frame of the 2-sphere \(\mathbb {S}^2_{(t,r)}\)  at \((t,r)\)  as follows
\(\hat{\phi }_a = r\phi _a = r[\nabla _a r(r^2\rho ) + e_{ab}\nabla ^b r(r^2\sigma )],\) 
we can define the following norms by the same way for the original field
\(|\hat{\Phi }|^2 := \hat{\phi }_a\hat{\phi }_b {g}^{ab}, \, |{\nabla }_L\hat{\Phi }|^2 := {\nabla }_L \hat{\phi }_a ({\nabla }_L \hat{\phi }_b) {g}^{ab} \hbox{ and } |{\nabla }_{\underline{L}}\hat{\Phi }|^2 := {\nabla }_{\underline{L}} \hat{\phi }_a ({\nabla }_{\underline{L}} \hat{\phi }_b) {g}^{ab},\) 
where the scalar product depends on the projected metric \({g}_{ab}\) .
Remark 3 Since we have
\(\hat{\phi }_a = r\phi _a = \nabla _{{\theta ^a}} r(r^2\rho ) + e_{\theta ^a\theta ^b}\nabla ^{{\theta ^b}} r(r^2\sigma ),\) 
the norms of the rescaled field \(\hat{\phi }_a\)  can be also defined on the unit 2-sphere \(\mathbb {S}^2\)  as
\(|\hat{\Phi }|_{\mathbb {S}^2}^2 = [\nabla _{{\theta ^a}} r(r^2\rho ) + e_{\theta ^a\theta ^b}\nabla ^{{\theta ^b}} r(r^2\sigma )] [\nabla _{{\theta ^a}} r(r^2\rho ) + e_{\theta ^a\theta ^b}\nabla ^{{\theta ^b}} r(r^2\sigma )] {g}_{\mathbb {S}^2}^{\theta ^a\theta ^b}.\) 
Clearly, we have
\(|\hat{\Phi }|_{\mathbb {S}^2}^2 &=& r^2[\nabla _{{\theta ^a}} ( r(r^2\rho )) + e_{\theta ^a\theta ^b}\nabla ^{{\theta ^b}}( r(r^2\sigma ))] [\nabla _{{\theta ^a}}(r(r^2\rho )) + e_{\theta ^a\theta ^b}\nabla ^{{\theta ^b}} (r(r^2\sigma ))] \frac{1}{r^2}{g}_{\mathbb {S}^2}^{\theta ^a\theta ^b}\\&=& r^2|\hat{\Phi }|.\) 

The rescaled Equation (REF ) leads to
\(&&4\left( {\nabla }_{\underline{L}}{\nabla }_L\hat{\phi }_a - FR^2{\Delta }_{\mathbb {S}^2}\hat{\phi }_a + (2MR+1)FR^2\hat{\phi }_a \right) ({\nabla }_{L}\hat{\phi }_b){g}^{ab} \\&&+ 4\left( {\nabla }_{\underline{L}}{\nabla }_L\hat{\phi }_a - FR^2{\Delta }_{\mathbb {S}^2}\hat{\phi }_a + (2MR+1)FR^2\hat{\phi }_a \right) ({\nabla }_{\underline{L}}\hat{\phi }_b) {g}^{ab} = 0.\) 
By using \({\nabla }_{L}{g}^{ab} = {\nabla }_{\underline{L}}{g}^{ab}=0\)  we see that this equation is equivalent to
\(&&2{\nabla }_{\underline{L}} \left( |{\nabla }_L\hat{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\nonumber \\&&+2{\nabla }_{L} \left( |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2} \hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) = 0.\) 
If we use the norms \(|.|_{\mathbb {S}^2}\)  on the unit 2-sphere \(\mathbb {S}^2\)  we will get the same equality as (REF ) due to
\(L(r^2)+\underline{L}(r^2) = 0.\) 
Raising the index of (REF ) by \(\hat{g}^{-1} = 4r^2F^{-1}\partial _u\partial _v - g^{-1}_{\mathbb {S}^2}\) , we get
\(&&8r^2F^{-1}{\nabla }^{L} \left( |{\nabla }_L\hat{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\nonumber \\&&+8r^2F^{-1} {\nabla }^{\underline{L}} \left( |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2} \hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) = 0 .\) 
Equation (REF ) is also equivalent to
\(&&({{\nabla }}_L + {\nabla }_{\underline{L}}) \left( |{\nabla }_L\hat{\Phi }|^2 + |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + 2F|R{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + 2(2MR+1)R^2F|\hat{\Phi }|^2 \right) \\&&- ({\nabla }_L - {\nabla }_{\underline{L}}) \left( |{\nabla }_L\hat{\Phi }|^2 - |{\nabla }_{\underline{L}}\hat{\Phi }|^2 \right) = 0.\) 
Hence, in the coordinates \((t,\, r_*,\, \omega )\)  Equation (REF ) becomes
\(&&2{\nabla }_{\partial _t} \left( |{\nabla }_L\hat{\Phi }|^2 + |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + 2F|R{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + 2(2MR+1)R^2F|\hat{\Phi }|^2 \right)\nonumber \\&&+2 {\nabla }_{\partial _{r_*}} \left( |{\nabla }_L\hat{\Phi }|^2 - |{\nabla }_{\underline{L}}\hat{\Phi }|^2 \right) = 0.\) 
Since \(\hat{g}^{-1} = r^2F^{-1}(\partial _t^2 - \partial _{r_*}^2) - g^{-1}_{\mathbb {S}^2}\) , we obtain that
\(&&2r^2F^{-1}{\nabla }^{\partial _t} \left( |{\nabla }_L\hat{\Phi }|^2 + |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + 2F|R{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + 2(2MR+1)R^2F|\hat{\Phi }|^2 \right)\nonumber \\&&+2r^2F^{-1} {\nabla }^{\partial _{r_*}} \left( |{\nabla }_L\hat{\Phi }|^2 - |{\nabla }_{\underline{L}}\hat{\Phi }|^2 \right) = 0.\) 
Equations (REF ) and (REF ) are the expressions of the rescaled tensorial conservation law for (REF ) in the coordinates \((u,\, v,\, \omega )\)  and \((t,\, r_*,\omega )\)  respectively.
We can re-write (REF ) and (REF ) by a generalized form as
\(f(r,F){\nabla }^a C(\hat{\Phi }) = 0,\) 
where \(\partial _a \in \left\lbrace  L,\, \underline{L}, \, \partial _t,\, \partial _{r_*} \right\rbrace \)  and \(f(r,F)=8r^2F^{-1}\)  in coordinates \((u,\,v,\,\omega )\)  and \(f(r,F)=2r^2F^{-1}\)  in coordinates \((t,\, r_*,\, \omega )\) .
Suppose \(\hat{\phi }_a\)  is a solution of the rescaled tensorial Equation (REF ) with the smooth compactly supported initial data. By integrating the conservation law (REF ) of \(\hat{\phi }_a\)  on the domain \(O\)  bounded by the hypersurfaces \(\Sigma _0\)  and \(\Sigma _t\)  in \((\bar{\mathcal {B}}_I, \hat{g})\)  and by using Stokes's formula we get
\(\int _O f(r,F){\nabla }^a C(\hat{\Phi }) \mathrm {dVol}_{\hat{g}} = \int _{\partial O} C(\hat{\Phi }) f(r,F)\partial _a { \begin{picture}(.833,.8)\put (.15,.08){\line (1,0){.35}}\put (.5,.08){\line (0,1){.5}}\end{picture}}\mathrm {dVol}_{\hat{g}},\) 
where \(\mathrm {dVol}_{\hat{g}} = R^2F\mathrm {d}t \mathrm {d}r_* \mathrm {d}\mathbb {S}^2\) . This implies that
\(\mathcal {E}_{\Sigma _0}(\hat{\Phi }) = \mathcal {E}_{\Sigma _t}(\hat{\Phi }),\) 
where
\(\mathcal {E}_{\Sigma _0}(\hat{\Phi }) = 4\int _{\Sigma _0}\left( |{\nabla }_{\partial _{t}}\hat{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) \mathrm {d}r_*\mathrm {d}\mathbb {S}^2.\) 
and
\(\mathcal {E}_{\Sigma _t}(\hat{\Phi }) = 4\int _{\Sigma _t}\left( |{\nabla }_{\partial _{t}}\hat{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) \mathrm {d}r_*\mathrm {d}\mathbb {S}^2.\) 
We recall that \(\Omega \)  is the domain bounded by the following five hypersurfaces
\(\bar{\Sigma }_0 = \left\lbrace  t=0 \right\rbrace ,\, \mathcal {H}^+_{2u_T} = \left\lbrace u=2u_T,v\le v_T \right\rbrace ,\,\mathcal {I}^+_{2v_T}= \left\lbrace v=2v_T,u \le u_T \right\rbrace ,\) 
\(\mathcal {H}_{u_T}= \left\lbrace u=u_T, v_T \le v \le 2v_T \right\rbrace ,\, \mathcal {I}_{v_T}= \left\lbrace v=v_T,u_T \le u \le 2u_T \right\rbrace .\) 
By integrating the conservation law (REF ) of \(\hat{\phi }_a\)  on the domain \(\Omega \)  in \((\bar{\mathcal {B}_I},\hat{g})\)  and by using Stokes's formula we get
\(\int _\Omega f(r,F){\nabla }^a C(\hat{\Phi }) \mathrm {dVol}_{\hat{g}} = \int _{\partial \Omega } C(\hat{\Phi }) f(r,F)\partial _a { \begin{picture}(.833,.8)\put (.15,.08){\line (1,0){.35}}\put (.5,.08){\line (0,1){.5}}\end{picture}}\mathrm {dVol}_{\hat{g}},\) 
where \(\mathrm {dVol}_{\hat{g}} = \dfrac{R^2F}{2}\mathrm {d}u \mathrm {d}v \mathrm {d}\mathbb {S}^2 = R^2F\mathrm {d}t \mathrm {d}r_* \mathrm {d}\mathbb {S}^2\) . This implies that
\(\mathcal {E}_{\mathcal {H}^+_{2u_T}}(\hat{\Phi }) + \mathcal {E}_{\mathcal {I}_{v_T}}(\hat{\Phi }) + \mathcal {E}_{\mathcal {H}_{u_T}}(\hat{\Phi }) + \mathcal {E}_{\mathcal {I}_{2v_T}^+}(\hat{\Phi }) = \mathcal {E}_{\bar{\Sigma }_0}(\hat{\Phi }),\) 
where
\(\mathcal {E}_{\bar{\Sigma }_0}(\hat{\Phi }) = 4\int _{\bar{\Sigma }_0}\left( |{\nabla }_{\partial _{t}}\hat{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) \mathrm {d}r_*\mathrm {d}\mathbb {S}^2,\) 
\(\mathcal {E}_{\mathcal {I}^+_{2v_T}}(\hat{\Phi }) = 4\int _{-\infty }^{u_T} \int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\mathrm {d}u \mathrm {d}\mathbb {S}^2,\) 
\(\mathcal {E}_{\mathcal {I}_{v_T}}(\hat{\Phi }) = 4\int _{u_T}^{2u_T}\int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\mathrm {d}u \mathrm {d}\mathbb {S}^2.\) 
\(\mathcal {E}_{\mathcal {H}^+_{2u_T}}(\hat{\Phi }) = 4\int ^{v_T}_{-\infty } \int _{\mathbb {S}^2} \left( |{\nabla }_L\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\mathrm {d}v \mathrm {d}\mathbb {S}^2, \) 
\(\mathcal {E}_{\mathcal {H}_{u_T}}(\hat{\Phi }) = 4\int ^{2v_T}_{v_T} \int _{\mathbb {S}^2} \left( |{\nabla }_L\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right)\mathrm {d}v \mathrm {d}\mathbb {S}^2. \) 
Remark 4 We observe that the hypersurfaces \(\mathcal {H}_{u_T}\)  and \(\mathcal {I}_{v_T}\)  will tend to the future horizon \(\mathfrak {H}^+\)  and future null infinity \({I}^+\)  as \(u_T\)  and \(v_T\)  (as well as \(T\) ) tend to infinity respectively. Therefore, the energy on the future horizon and null infinity are defined as follows
\(\mathcal {E}_{\mathfrak {H}_T^+}(\hat{\Phi }) = 4\int _{\mathfrak {H}^+_T} |{\nabla }_L\hat{\Phi }|_{\mathfrak {H}_T^+}|^2 \mathrm {d}v \mathrm {d}\mathbb {S}^2,\) 
\(\mathcal {E}_{{I}_T^+}(\hat{\Phi }) = 4\int _{{I}^+_T} |{\nabla }_{\underline{L}}\hat{\Phi }|_{{I}_T^+}|^2 \mathrm {d}u \mathrm {d}\mathbb {S}^2.\) 

As a direct consequence of the energy equality (REF ) we have the following result.
Proposition 2 Consider the smooth and compactly supported initial data on \(\Sigma _0\) , we can define the energy fluxes of the rescaled solution \(\hat{\phi }_a\)  of (REF ) across the null conformal boundary \(\mathfrak {H}^+\cup {I}^+\)  by \({\mathcal {E}}_{{I}^+}(\hat{\Phi }) + {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\Phi }) := \lim _{T\rightarrow \infty }{\mathcal {E}}_{{I}_T^+}(\hat{\Phi }) + {\mathcal {E}}_{\mathfrak {H}_T^+}(\hat{\Phi })\) . Moreover, we have
\({\mathcal {E}}_{{I}^+}(\hat{\Phi }) + {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\Phi }) \le {\mathcal {E}}_{\Sigma _0}(\hat{\Phi }),\) 
the equality holds if and only if
\(\lim _{u_T,v_T\rightarrow \infty } \left( {\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\Phi }) + {\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\Phi })\right) = 0.\) 

By the same manner for \(\hat{\phi }_a\) , we can also establish the energy fluxes of \(\hat{\underline{\phi }}_a\)  and then we obtain the energy equality as follows.
Proposition 3 Consider the smooth and compactly supported initial data on \(\Sigma _0\) , we can define the energy fluxes of the rescaled solution \(\hat{\underline{\phi }}_a\)  of (REF ) across the null conformal boundary \(\mathfrak {H}^+\cup {I}^+\)  by \({\mathcal {E}}_{{I}^+}(\hat{\underline{\Phi }}) + {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\underline{\Phi }}) := \lim _{T\rightarrow \infty }{\mathcal {E}}_{{I}_T^+}(\hat{\underline{\Phi }}) + {\mathcal {E}}_{\mathfrak {H}_T^+}(\hat{\underline{\Phi }})\) . Moreover, we have
\({\mathcal {E}}_{{I}^+}(\hat{\underline{\Phi }}) + {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\underline{\Phi }}) \le {\mathcal {E}}_{\Sigma _0}(\hat{\underline{\Phi }}),\) 
the equality holds if and only if
\(\lim _{u_T,v_T\rightarrow \infty } \left( {\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\underline{\Phi }}) + {\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\underline{\Phi }})\right) = 0.\) 
Here we define
\(\mathcal {E}_{\Sigma _0}(\hat{\underline{\Phi }}) = 4\int _{\Sigma _0}\left( |{\nabla }_{\partial _{t}}\hat{\underline{\Phi }}|^2 + |{\nabla }_{\partial _{r_*}}\hat{\underline{\Phi }}|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\underline{\Phi }}|^2 + (2MR+1)R^2F|\hat{\underline{\Phi }}|^2 \right) \mathrm {d}r_*\mathrm {d}\mathbb {S}^2.\) 
\(\mathcal {E}_{\mathcal {I}_{v_T}}(\hat{\underline{\Phi }}) = 4\int _{u_T}^{2u_T}\int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}\hat{\underline{\Phi }}|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\underline{\Phi }}|^2 + (2MR+1)R^2F|\hat{\underline{\Phi }}|^2 \right)\mathrm {d}u \mathrm {d}\mathbb {S}^2.\) 
\(\mathcal {E}_{\mathcal {H}_{u_T}}(\hat{\underline{\Phi }}) = 4\int ^{2v_T}_{v_T} \int _{\mathbb {S}^2} \left( |{\nabla }_L\hat{\underline{\Phi }}|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\underline{\Phi }}|^2 + (2MR+1)R^2F|\hat{\underline{\Phi }}|^2 \right)\mathrm {d}v \mathrm {d}\mathbb {S}^2. \) 


Tensorial field space of initial data and Cauchy problem
Since the energy of the rescaled field \(\hat{\phi }_a\)  across the hypersurface \(\Sigma _0\)  is given by (REF ), we define the finite energy space as follows.
Definition 1 We denote by \(\mathcal {H}\)  the completion of \(\mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0})\times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0})\)  in the norm
\(\left\Vert (\hat{\Phi }_0,\hat{\Phi }_1) \right\Vert _{\mathcal {H}} = 2\left( \int _{\Sigma _0}\left( |{\nabla }_{\partial _{t}}\hat{\Phi }|^2 + |{\nabla }_{\partial _{r_*}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) \mathrm {d}r_*\mathrm {d}\mathbb {S}^2 \right)^{1/2}.\) 

Before stating the theorem of the well-posedness of Cauchy problem, we need the following definition of the Sobolev spaces for tensorial fields that are defined on open sets (see Definition 2 [2]} for the same definition for scalar fields):
Definition 2 Let \(s \in [0, +\infty )\) , a tensorial field \(u_a\)  on \(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {B}_I}\)  is said to belong to \(H^s_{loc}(\Lambda ^1(\mathbb {S}^2)|_{\bar{\mathcal {B}_I})}\)  if for any
local chart \((\Omega , \zeta )\) , such that \(\Omega \subset \mathcal {B}_I\)  is an open set with smooth compact boundary in \(\bar{\mathcal {B}_I}\)  (note that this excludes neighbourhoods of either \(i^\pm \)  or \(i^0\)  but allows open sets whose boundary contains parts of the conformal boundary) and \(\zeta \)  is a smooth diffeomorphism from \(\Omega \)  onto a bounded open set \(U \subset \mathbb {R}^4\)  with smooth compact boundary, we have \(u_a \circ \zeta ^{-1} \in H^s(\Lambda ^1(\mathbb {S}^2)|_U)\) .

By Leray's theorem for the hyperbolic differential equations and the energy identities (REF ) and (REF ), we can establish the well-posedness of Cauchy problem for Equation (REF ) and the rescaled Equation equation (REF ) as follows
Proposition 4 The Cauchy problem for (REF ) on \(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {B}_I})\)  (and for (REF ) on \(\Lambda ^1(\mathbb {S}^2)|_{\bar{\mathcal {B}_I}}\) ) is well-posed in \(\mathcal {H}\) . This means that for any \((\hat{\phi }_{a0},\hat{\phi }_{a1}) \in \mathcal {H}\) , there exists a unique solution \(\phi _a \in \mathcal {D}^{\prime }(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {B}_I)}\)  of (REF ) such that
\((r\phi _a,r{\nabla }_{\partial _t}\phi _a) \in \mathcal {C}(\mathbb {R}_t;\, \mathcal {H}): \, r\phi _a|_{t=0}=\hat{\phi }_{a0}; \, r{\nabla }_{\partial _t}\phi _a|_{t=0} = \hat{\phi }_{a1}.\) 
Moreover, \(\hat{\phi }_a=r\phi _a\)  belongs to \(H^1_{loc}(\Lambda ^1(\mathbb {S}^2)|_{\bar{{\mathcal {B}}_I}})\) .


Energy identity up to \(i^+\)  and trace operator
In this section, we will show that \(\lim _{u_T,v_T\rightarrow \infty } \left( {\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\Phi }) + {\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\Phi })\right) = 0\)  and then we can obtain the energy equality
\({\mathcal {E}}_{\Sigma _0}(\hat{\Phi }) = {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\Phi }) + {\mathcal {E}}_{{I}^+}(\hat{\Phi }).\) 
We shall use the following results about the energy and pointwise decays of the tensorial field \(\phi _a\)  which satisfies Equation (REF ). These results are a direct consequence of Theorems 4.2 and Lemma 5.8 in [3]}).
Theorem 1 
There exist a positive number \((r_{FH})_*\)  and positive constants \(C\)  and \(D\)  depending only on \(M\)  and \((r_{FH})_*\)  such that the following assertions hold
i)
(Energy Decay). Let \(\phi _a\)  be a smooth tensorial solution to the Fackerell-Ipser Equations (REF ) on \(\left\lbrace  u \ge u_0, \, v\ge v_0 \right\rbrace \) . We have the following decay energy for the original field \(\phi _a\)  on \(\left\lbrace u=u_T,v\ge v_T \right\rbrace  \cup \left\lbrace  u\ge u_T,v=v_T \right\rbrace \)  (where \(u_T\ge u_0\)  and \(v_T= u_T +2(r_{FH})_*\) ):
\(\mathcal {E}_{u=u_T,v\ge v_T}(\Phi ) + \mathcal {E}_{u\ge u_T,v=v_T}(\Phi ) \le C u^{-2}.\) 
ii)
(Pointwise Decay). In the region \( \left\lbrace  r_*\le (r_{FH})_* \right\rbrace  \cap \left\lbrace  u \ge u_0, \, v\ge v_0 \right\rbrace \)  the solution \(\phi _a\)  of (REF ) decays as \(v^{-1}\) . In the region \(\left\lbrace  r_*\ge (r_{FH})_* \right\rbrace  \cap \left\lbrace  u \ge u_0,\, v \ge v_0 \right\rbrace \) , it decays as \(r^{-1}(|u|+1)^{-1/2}\)  or \(r^{-1/2}(|u|+1)^{-1}\)  i.e
\(|\Phi |^2 \le \frac{D}{r(|u|+1)^{1/2}} \hbox{ and } \frac{D}{r^{1/2}(|u|+1)}.\) 


Remark 5 Assertion i) is an exact consequence of Lemma 5.8 in [3]}. On the other hand, Assertion ii) is obtained by combining Remark REF  and Theorem 4.2 in [3]}.

Using these decay results we prove the following theorem.
Theorem 2 Let \(\hat{\phi }_a\)  (resp. \(\hat{\underline{\phi }}_a\) )be the solution of the rescaled tensorial Fackerell-Ipser Equation (REF ) (resp. (REF )). The energy of \(\hat{\phi }_a\)  (resp. \(\hat{\underline{\phi }}_a\) ) across the hypersurface \(\mathcal {H}_T\)  tends to zero as \(T\)  tends to infinity
\(\lim _{u_T,v_T\rightarrow \infty } \left( {\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\Phi }) + {\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\Phi })\right) = 0\) 
\(( \hbox{resp. } \lim _{u_T,v_T\rightarrow \infty } \left( {\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\underline{\Phi }}) + {\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\underline{\Phi }})\right) = 0.)\) 
As a consequence, the equality of the energies remains true
\({\mathcal {E}}_{\Sigma _0}(\hat{\Phi }) = {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\Phi }) + {\mathcal {E}}_{{I}^+}(\hat{\Phi }),\) 
\((\hbox{resp. } {\mathcal {E}}_{\Sigma _0}(\hat{\underline{\Phi }}) = {\mathcal {E}}_{\mathfrak {H}^+}(\hat{\underline{\Phi }}) + {\mathcal {E}}_{{I}^+}(\hat{\underline{\Phi }}).)\) 

In the region \(\left\lbrace  r_*\ge (r_{FH})_* \right\rbrace \) , we have
\({\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\Phi }) &=& 4\int _{v_T}^{2v_T} \int _{\mathbb {S}^2} \left( |{\nabla }_L\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) \mathrm {d}v \mathrm {d}\mathbb {S}^2\\&=& 4\int _{v_T}^{2v_T} \int _{\mathbb {S}^2} \left( r^2|{\nabla }_L{\Phi }|^2 + F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + F^2|\Phi |^2 + (2MR+1)F|\Phi |^2 \right) \mathrm {d}v \mathrm {d}\mathbb {S}^2\\&=& 4\int _{v_T}^{2v_T} \int _{\mathbb {S}^2} \left( |{\nabla }_L{\Phi }|^2 + F|R{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + FR^2|\Phi |^2 \right) r^2\mathrm {d}v\mathrm {d}\mathbb {S}^2\\&&+4\int _{v_T}^{2v_T} \int _{\mathbb {S}^2} \left( (F^2-F)|\Phi |^2 + (2MR+1)F|{\Phi }|^2 \right)\mathrm {d}v \mathrm {d}\mathbb {S}^2\\&\lesssim & \mathcal {E}_{\mathcal {I}_{v_T}}(\Phi ) + \int _{v_T}^{2v_T} \int _{\mathbb {S}^2}|\Phi |^2 \mathrm {d}v \mathrm {d}\mathbb {S}^2.\) 
Using the decay results in Theorem REF  we get
\(\mathcal {E}_{\mathcal {I}_{v_T}}(\hat{\Phi })&\lesssim & Cu_T^{-2} + \int _{v_T}^{2v_T} \frac{4\pi D}{r(u_T+1)^2} \mathrm {d}v \\&\lesssim & Cu_T^{-2} + \frac{v_T}{(r_{FH})_*(u_T+1)^2} = cu_T^{-2} + 4\pi D\frac{u_T + 2(r_{FH})_*}{(r_{FH})_*(u_T+1)^2} \longrightarrow 0 \hbox{ as } u_T \rightarrow \infty .\) 
On the other hand, in the region \(\left\lbrace  r_*\le (r_{FH})_* \right\rbrace \)  we have
\({\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\Phi }) &=& 4\int _{u_T}^{2u_T} \int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}\hat{\Phi }|^2 + R^2F|{\nabla }_{\mathbb {S}^2}\hat{\Phi }|^2 + (2MR+1)R^2F|\hat{\Phi }|^2 \right) \mathrm {d}u \mathrm {d}\mathbb {S}^2\\&=& 4\int _{u_T}^{2u_T} \int _{\mathbb {S}^2} \left( r^2|{\nabla }_{\underline{L}}{\Phi }|^2 + F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + F^2|\Phi |^2 + (2MR+1)F|{\phi }|^2 \right)\mathrm {d}u \mathrm {d}\mathbb {S}^2\\&=& 4\int _{u_T}^{2u_T} \int _{\mathbb {S}^2} \left( |{\nabla }_{\underline{L}}{\Phi }|^2 + F|{\nabla }_{\mathbb {S}^2}{\Phi }|^2 + FR^2|\Phi |^2 \right) r^2\mathrm {d}u\mathrm {d}\mathbb {S}^2\\&&+4\int _{u_T}^{2u_T} \int _{\mathbb {S}^2} \left( (F^2-F)|\Phi |^2 + (2MR+1)F|{\Phi }|^2 \right)\mathrm {d}u \mathrm {d}\mathbb {S}^2\\&\lesssim & \mathcal {E}_{\mathcal {H}_{u_T}}(\Phi ) + \int _{u_T}^\infty \int _{\mathbb {S}^2}|\Phi |^2 \mathrm {d}v \mathrm {d}\mathbb {S}^2.\) 
Using again Theorem REF  we get
\({\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\Phi })&\lesssim & Cu_T^{-2} + \int _{u_T}^{2u_T} \frac{D}{v_T^2} \mathrm {d}u \\&\lesssim & Cu_T^{-2} + \frac{u_T}{v_T^2} = Cu_T^{-2} + 4\pi D\frac{v_T-2(r_{FH})_*}{v_T^2} \longrightarrow 0 \hbox{ as } {u_T,\,v_T} \rightarrow \infty .\) 
Therefore we establish that
\(\lim _{u_T,v_T\rightarrow \infty } \left( {\mathcal {E}}_{\mathcal {H}_{u_T}}(\hat{\Phi }) + {\mathcal {E}}_{\mathcal {I}_{v_T}}(\hat{\Phi })\right) = 0.\) 
Together with (REF ), this gives the equality of the energies (REF ) for \(\hat{\phi }_a\) . The equality energy for \(\hat{\underline{\phi }}_a\)  is established in the same way. Our proof is then completed.

The energy equality leads us to define the trace operator on the conformal boundary as follows.
Definition 3 (Trace operator). Let \((\hat{\phi }_{a0}, \hat{\phi }_{a1}) \in \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0})\times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0})\) . Consider the solution of the rescaled Equations (REF ), and let \(\hat{\phi }_a \in \mathcal {C}^\infty (\Lambda ^1(\mathbb {S}^2)|_{\bar{\mathcal {B}_I}})\)  such that
\(\hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0}} = \hat{\phi }_{a0}, \, {\nabla }_t \hat{\phi }_{a}|_{\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0}} = \hat{\phi }_{a1}.\) 
We define the trace operator \(\mathcal {T}^+\)  from \(\mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0})\times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0})\)  to \(\mathcal {C}^\infty (\Lambda ^1(\mathbb {S}^2)|_{\mathfrak {H}^+})\times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{{I}^+})\)  by
\(\mathcal {T}^+(\hat{\phi }_{a0}, \hat{\phi }_{a1}) = (\hat{\phi }_a|_{\mathfrak {H}^+}, \hat{\phi }_a|_{{I}^+}).\) 

Then we can extend the function space for scattering data by density as in the following definition.
Definition 4 The function space for scattering data \(\mathcal {H}^+\)  is the completion of \(\mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\mathfrak {H}^+}) \times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{{I}^+})\)  in the norm
\(\left\Vert  (\xi _a,\zeta _a) \right\Vert _{\mathcal {H}^+} = \left(\int _{\mathfrak {H}^+} |{\nabla }_L \xi |^2 \mathrm {d}v\mathrm {d}^2\omega + \int _{{I}^+} |{\nabla }_{\underline{L}} \zeta |^2 \mathrm {d}u \mathrm {d}^2\omega \right)^{1/2}.\) 
This means that
\(\mathcal {H}^+ \simeq \dot{H}^1(\mathbb {R}_v; \, L^2(\Lambda ^1(\mathbb {S}^2))) \times \dot{H}^1(\mathbb {R}_u; \, L^2(\Lambda ^1(\mathbb {S}^2))).\) 

As a direct consequence of the equality energy (REF ), we have the following theorem.
Theorem 3 
The trace operator extends uniquely as a bounded linear map from \(\mathcal {H}\)  to \(\mathcal {H}^+\) . The extended operator is a partial isometry i.e for any \((\hat{\phi }_{a0},\hat{\phi }_{a1}) \in \mathcal {H}\) ,
\(\left\Vert  \mathcal {T}^+(\hat{\phi }_{a0},\hat{\phi }_{a1}) \right\Vert _{\mathcal {H}^+} = \left\Vert  (\hat{\phi }_{a0},\hat{\phi }_{a1}) \right\Vert _{\mathcal {H}}.\) 


Conformal scattering theory
Generalisation of L. Hörmander's result for tensorial wave equations
Hörmander [6]} proved the well-posedness of the Goursat problem for the second order scalar wave equations with regular first order potentials in the spatially compact spacetime. Nicolas [7]} extended the results of Hörmander with very slightly more regular metric and potential, precisely a \(\mathcal {C}^1\)  metric and a potential with continuous coefficients of the first order terms and locally \(L^\infty \)  coefficients for the terms of order 0.
Here we will solve the Goursat problem for the tensorial wave equations with regular first order tensorial potentials. More precisely, we will show how we can apply the results of Hörmander and Nicolas for the tensorial wave Equation (REF ) (or (REF )) with the smooth compactly supported initial data on the conformal boundary i.e, \((\xi _a, \zeta _a) \in \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\mathfrak {H}^+}) \times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{{I}^+})\)  in Schwarzschild background.
By the same manner in Appendix B [2]}, the conformal compactification domain \(\bar{\mathcal {B}}_I\)  is embedded partly in a cylindrical globally hyperbolic spacetime by the following technique: let \(\mathcal {S}\)  be a spacelike hypersurface on \(\bar{\mathcal {B}}_I\)  whose intersection with the horizon is the crossing sphere and which crosses \({I}^+\)  strictly in the past of the support of the data. We cut off the future \(\mathcal {V}\)  of a point in \(\bar{\mathcal {B}}_I\)  lying in the future of the past of the support of the Goursat data and get a spacetime denoted by \(\mathfrak {M}\) . Then we extend \(\mathfrak {M}\)  as a cylindrical globally hyperbolic spacetime \((\mathbb {R}_t\times S^3, \mathfrak {g})\) . The conformal boundary \(\mathfrak {H}^+\cup {I}^+\)  is extended inside \(\mathcal {I}^+(\mathcal {S})/\mathcal {V}\)  as a null hypersurface \(\mathcal {C}\) , that is the graph of a Lipschitz function over \(S^3\)  and the data by zero on the rest of the extended hypersurface.
<FIGURE>By following Remark REF  we have the decomposition of the rescaled tensorial Equation (REF ) on the basic frame \(({\nabla }_a,{\nabla }_b)\)  of the 2-sphere \(\mathbb {S}^2_{(t,r)}\)  obtained as follows
\(P\hat{\Phi } + L_1 \hat{\Phi } = 0 \, ,\) 
with
\(P= \left(\begin{matrix}\Box _{g} &&0\\0&&\Box _{g}\end{matrix} \right)\) 
is the \(2\times 2\) -matrix diagram with \(\Box _g = \frac{1}{F}L\underline{L} - {\Delta }\) ,
\(\hat{\Phi } = \left( \begin{matrix} \hat{\phi }_1\\ \hat{\phi }_2\end{matrix} \right),\) 
where \(\hat{\phi }_i \, (i=1,2)\)  are the components of \(\hat{\phi }_a\)  decomposed on the basic frame \(({\nabla }_a,{\nabla }_b)\)  of \(\Lambda ^1(\mathcal {B})\)  and
\(L_1 = \left( \begin{matrix} L_1^{11}&&L_1^{12}\\ L_1^{21}&& L_1^{22} \end{matrix} \right)\) 
is the \(2\times 2\) -matrix where the components are the operators that have the smooth coefficients
\(L_1^{ij} = b_0^{ij}\partial _t + b_a^{ij}\partial _a + c^{ij}.\) 
In the spacetime \((\mathbb {R}_t\times S^3, \mathfrak {g})\) , Equation (REF ) becomes
\(P_{\mathfrak {g}}\hat{\Phi } + L_1 \hat{\Phi } = 0 \, ,\) 
where
\(P_{\mathfrak {g}}= \left(\begin{matrix}\Box _{\mathfrak {g}} &&0\\0&&\Box _{\mathfrak {g}}\end{matrix} \right).\) 
If we define the norm of the vector by the maximal element's norm, we have that
\( 0 = \sum _{i=1}^2 \partial _t \bar{\hat{\phi }}_i \left( P\hat{\Phi }+ L_1 \hat{\Phi }\right)_i = \sum _{i=1}^2 \partial _t \bar{\hat{\phi }}_i \left( P\hat{\phi }_i + (L_1\hat{\Phi })_i \right)\) 
and
\(\sum _{i=1}^2 \partial _t\bar{\hat{\phi }}_i (L_1\hat{\Phi })_i \le \left\Vert  \partial _t\hat{\Phi }\right\Vert  \left\Vert L_1\hat{\Phi }\right\Vert  \le \left\Vert  \partial _t\hat{\Phi }\right\Vert  \left( \left\Vert  \nabla \hat{\Phi }\right\Vert  + \left\Vert  \partial _t\hat{\Phi }\right\Vert  + \left|\hat{\Phi }\right\Vert  \right).\) 
The above estimates are used to control the energies for the matrix wave equation (REF ) by the same method for the scalar wave equation in [7]}. In addition the metric \(\mathfrak {g}\)  and the coefficients in \(L_1\)  operator satisfy the conditions \({\bf H}_1\)  and \({\bf H}_2\)  in [7]}. Therefore, The Goursat problem for the system (REF ) is solved by the same manner for the scalar wave equation with regular first order potential in [7]}.
By local uniqueness and causality, using in particular the fact that as a consequence of the
finite propagation speed, the solution \(\Phi \)  vanishes in \(\mathcal {I}^+(\mathcal {S})/\bar{\mathcal {B}}_I\) , the Goursat problem of the equation (REF ) has a unique smooth solution in the future of \(\mathcal {S}\) , that is the restriction of \(\Phi \)  to \(\mathfrak {M}\) . Therefore, we have established the well-posedness of the Goursat problem of (REF ), hence (REF ) in the following theorem.
Theorem 4 
With the initial data \((\xi , \, \zeta ) \in {\mathcal {C}}_0^{\infty }(\mathfrak {H}^+)\times {\mathcal {C}}_0^\infty ({I}^+) \)  where \(\xi =(\xi _1,\, \xi _2)\)  and \(\zeta = (\zeta _1,\, \zeta _2)\) , Equation (REF ) has a unique solution \(\hat{\Phi } = (\hat{\phi }_1,\, \hat{\phi }_2)\)  satisfying
\(\hat{\phi }_i \in {\mathcal {C}}(\mathbb {R};\, H^1(\mathcal {S}))\cap \mathcal {C}^1(\mathbb {R}; \, L^2(\mathcal {S})),\) 
for all \(i = 1,\, 2.\)
In the term of tensor: with the initial data \((\xi _a, \zeta _a) \in \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\mathfrak {H}^+}) \times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{{I}^+})\) , the rescaled tensorial equation (REF ) has a unique solution \(\hat{\phi }_{a}\)  such that
\(\hat{\phi }_{a} \in {\mathcal {C}}(\mathbb {R};\, H^1(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}))\cap \mathcal {C}^1(\mathbb {R}; \, L^2(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}})).\) 

Remark 6 We have solved partly the Goursat problem for the tensorial Equation (REF ) in the future \(\mathcal {I}^+(\mathcal {S})\) , the problem will be solved completely by extending this solution down to \(\Sigma _0\) . This will be done in the next section.


Goursat problem and conformal scattering operator
In order to obain the conformal scattering operator, we need to show that the trace operator is subjective. This corresponds to solve the Goursat problem for the rescaled Equation (REF ) with the initial data on the conformal boundaries \(\mathfrak {H}^+\cup {I}^+\)  (resp. \(\mathfrak {H}^-\cup {I}^-\) ) in the conformal compactification domain \(\bar{\mathcal {B}}_I\) . In the previous section, we have solved the Goursat problem in the future of the spacelike hypersurface \(\mathcal {S}\) .
The following theorem gives the full solution of the problem in \(\bar{\mathcal {B}}_I\) .
Theorem 5 
The Goursat problem of the rescaled tensorial Equation (REF ) is well-posed. This means that for the initial data \((\xi ,\zeta )\in C_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\mathfrak {H}^+})\times C_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{{I}^+})\) , there exists a unique solution of (REF ) satisfying
\((\hat{\phi }_a,{\nabla }_t\hat{\phi }_a)\in \mathcal {C}(\mathbb {R}_t; \, \mathcal {H}) \hbox{ and } \mathcal {T}^+(\hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0}}, {\nabla }_t \hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0}}) = (\xi _a,\zeta _a).\) 

Following Theorem REF  there exists a unique solution \(\hat{\phi }_a\)  of (REF )
which satisfies the following properties.
\(\bullet \)
\(\hat{\phi }_a \in H^1(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {I}^+(\mathcal {S}}))\) , where \(\mathcal {I}^+(\mathcal {S})\)  is the causal future of \(\mathcal {S}\)  in \(\bar{\mathcal {B}_I}\) . Since the support of the initial data is compact,the solution vanishes in the neigbourhood of \(i^+\) . Then we do not need to distinguish between \(H^1(\mathcal {I}^+(\mathcal {S}))\)  and \(H^1_{loc}(\mathcal {I}^+(\mathcal {S}))\) .
\(\bullet \)
\(\hat{\phi }_a\)  is continuous in \(\tau \)  with values in \(H^1\)  of the slices and \(C^1\)  in \(\tau \)  with values in \(L^2\)  of the slices.
\(\bullet \)
\(\hat{\phi }_a|_{{I}^+} = \zeta _a, \, \hat{\phi }_a|_{\mathfrak {H}^+}= \xi _a\) .

<FIGURE>We need to extend the solution down to \(\Sigma _0\)  in a manner that avoids the singularity at \(i_0\) .
Since the hypersurface \(\mathcal {S}\)  intersects the horizon at the crossing sphere and intersects \({I}^+\)  strictly in the past of the support of the data, we have that the restriction of \(\hat{\phi }_a\)  to \(\mathcal {S}\)  is in \(H^1(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}})\)  and its trace on \(\mathcal {S}\cap {I}^+\)  is also the trace of \(\zeta _a\)  on \(\mathcal {S}\cap {I}^+\) . Therefore, \(\hat{\phi }_a|_{\Lambda ^1(\mathcal {S}^2)|_{\mathcal {S}}}\)  can be approached by a sequence \(\left\lbrace \hat{\phi }^n_{a0,\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}} \right\rbrace _{n\in \mathbb {N}}\)  of the smooth tensorial fields on \(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}\)  supported away from \({I}^+\)  that converges towards \(\hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}}\)  in \(H^1(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}})\) . Moreover, \({\nabla }_{\partial _t}\hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}}\)  can be approached by a sequence \(\left\lbrace \hat{\phi }^n_{a1,\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}} \right\rbrace _{n\in \mathbb {N}}\)  of the smooth tensorial fields on \(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}\)  supported away from \({I}^+\)  that converges towards \({\nabla }_{\partial _t}\hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}}\)  in \(L^2(\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}})\) .
Consider \(\hat{\psi }_a^n\) , the smooth tensorial solution of (REF ) on \(\Lambda ^1(\mathbb {S}^2)|_{\bar{\mathcal {B}_I}}\)  with data \((\hat{\phi }^n_{a0,\Lambda ^1{\mathbb {S}^2}|_{\mathcal {S}}},\hat{\phi }^n_{a1,\Lambda ^1(\mathbb {S}^2)|_{\mathcal {S}}})\)  on \(\mathcal {S}\) . This tensorial solution vanishes in the neighbourhood of \(i_0\)  and we can establish energy estimates for \(\hat{\psi }_a^n\)  between \(\mathcal {S}\)  and \(\Sigma _0\)  as follows
\(\mathcal {E}^{\partial _t}_{\mathcal {S}}(\hat{\psi }_a^n) = \mathcal {E}^{\partial _t}_{\Sigma _0}(\hat{\psi }_a^n).\) 
By the same way we have energy identities between \(\mathcal {S}\)  and the hypersurfaces \(\Sigma _t \hbox{ for } t> 0\) . Therefore, the sequence \((\hat{\psi }_a^n,{\nabla }_{\partial _t}\hat{\psi }_a^n)\)  converges towards \((\hat{\psi }_a, {\nabla }_{\partial _t}\hat{\psi }_a)\)  in \(\mathcal {C}(\mathbb {R}_t,\mathcal {H})\) , where \(\hat{\psi }_a\)  is a solution of (REF ). By local uniqueness \(\hat{\psi }_a\)  coincides with \(\hat{\phi }_a\)  in the future of \(\mathcal {S}\) . Therefore, if we denote
\(\hat{\phi }_{a0} = \hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0}},\,\, \hat{\phi }_{a1} = {\nabla }_{\partial _t}\hat{\phi }_a|_{\Lambda ^1(\mathbb {S}^2)|_{\Sigma _0}},\) 
we have
\((\hat{\phi }_{a0},\hat{\phi }_{a1})\in \mathcal {H} \hbox{ and } (\xi _a,\zeta _a) = \mathcal {T}^+(\hat{\phi }_{a0},\hat{\phi }_{a1}).\) 
Therefore, the range of \(\mathcal {T}^+\)  contains \(\mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{\mathfrak {H}^+})\times \mathcal {C}_0^\infty (\Lambda ^1(\mathbb {S}^2)|_{{I}^+})\) .
Remark 7 Theorem REF  shows that the trace operator \(\mathcal {T}^+: \mathcal {H} \rightarrow \mathcal {H}^+\)  is subjective. Combining with Theorem REF  we obtain that the trace operator \(\mathcal {T}^+: \mathcal {H} \rightarrow \mathcal {H}^+\)  is an isometry. Similarly, we can establish the past trace operator \(\mathcal {T}^-\)  and the space \(\mathcal {H}^-\)  of past scattering data on the past horizon and the past null infinity.

We now define the conformal scattering operator for the tensorial Fackerell-Ipser Equation (REF ) (which has the rescaled form (REF )) as follows.
Definition 5 The conformal scattering operator \(S\)  of Equation (REF ) is an isometry that maps the past scattering data to the future scattering data, i.e.
\(S:= \mathcal {T}^+\circ (\mathcal {T}^-)^{-1}.\) 

Remark 8
\(\bullet \)
By the same manner we can construct the conformal scattering operator for Equation (REF ).
\(\bullet \)
This work can be extended to construct a conformal scattering theory for the Fackerell-Ipser equations on the other symmetric spherical spacetimes (such as Reissner-Nordström-de Sitter back hole) can be done by the same way: first, by extending the work [3]} to obtain the decay results (where the results of Giorgi [13]} can be useful) and then using these results to the construction of the theory, where it remains useful to use the time-like Killing vector field \(\partial _t\)  to obtain the energy of the fields.
\(\bullet \)
Conformal scattering theories for the spin \(\pm 1\)  or \(\pm 2\)  Teukolsky equations can be seem also to construct firstly in Schwarzschild spacetime.
\(\bullet \)
The extension of the conformal scattering theory on the stationary and non symmetric spherical spacetimes such as Kerr spacetimes is unknown so far and remains an open problem. In Kerr spacetimes, the existence of the orbiting null geodesics and the fact that the vector field \(\partial _t\)  is no longer global time-like in the exterior of the black hole lead an issue that the energies on the Cauchy hypersurfaces \(\Sigma _T = \left\lbrace  t=T \right\rbrace \)  are not defined as usual.
\(\bullet \)
The peeling property of the rescaled Equations (REF ) is also an interesting question where the method can be done by the same manner for the linear scalar fields on Schwarzschild spacetimes in [14]} and extended to Kerr spacetimes in [15]}, [16]}.
